This paper introduces a new wheel-legged robot and develops motion controllers based on central pattern generators (CPGs) for the robot to navigate over a range of terrains. A transformable leg-wheel design is considered and characterized in terms of key locomotion characteristics as a function of the design. Kinematic analysis is conducted based on a generalized four-bar mechanism driven by a coaxial hub arrangement. The analysis is used to inform the design of a central pattern generator to control the robot by mapping oscillator states to wheel-leg trajectories and implementing differential steering within the oscillator network. Three oscillator models are used as the basis of the CPGs, and their performance is compared over a range of inputs. The CPG-based controller is used to drive the developed robot prototype on level ground and over obstacles. Additional simulated tests are performed for uneven terrain negotiation and obstacle climbing. Results demonstrate the effectiveness of CPG control in transformable wheel-legged robots.